Weight painting not moving model as expected
A weight of 1 (red) means, that your mesh will react 100% to the bone movement.
A weight of 0 (blue) means, that your mesh will not react at all.

So, if you want a more natural behavior, select the lower vertices of the head and assign a weight of 0.5 (green). These vertices will only be partially affected. You can go further and have several transition loops:
Assign it like this to the top bone.



And assign reverse weights to the bottom bone.